Analysis of External Ballistics of a Projectile of Calibre 155 mm

Rui Filipe da Silva Fonte-Boa

Academia Militar & Instituto Superior Técnico, Universidade de Lisboa, Portugal

December 7, 2014

Abstract

The objective of this work was to propose an external geometry for a firefighting projectile named

FIREND. The finite volume code Star − CCM+ R© is used to calculate forces and moments acting on the

projectile. The characteristics of the trajectory are computed using Mathematica R©. The PRODAS V 3 R©

software was used to verify and validate the models implemented.

Initially, the procedure adopted was to create a methodology for an existing ammunition that would

allow to be verified and validated with the available literature. An extensive aerodynamic characterization

has been done through CFD in order to establish the projectile in terms of forces and moments which depend

on the angle of attack and thus feed the script for 6-DOF computation. It was concluded that the locking

strap, the guiding strap and the Coriolis drift were negligible compared to the results obtained.

In the following phase six different geometries were tested, varying the lenght and the nose geometry.

It was defined for this projectile an initial velocity is 100 m/s and the weight 10 kg, concluding that the

total length would be 697.1 mm and hemispherical profile with the capacity to transport 7.5 dm3 inside.

Wind tests were conducted to evaluate the projectile stability in flight. The software and analysis process

developed in this work are available, being a contribution to the project FIREND.

Keywords: Firefighting projectile, projectile geometry, computation of aeroballistic coefficients, six

degree-of-freedom trajectory, dynamic stability analysis.

1 Introduction

The occurrence of forest fires is unavoidable. The

FIREND project arose in 2005 [1] with the ultimate

aim of providing an alternative and additional way

to forest fire fighting, having suffered its last up-

grade in 2013 [2]. The projectile calibre changed to

155 mm. The main reason that justified this change

was to increase the volume of firefighting agent to be

transported inside, which happened to be three times

higher.

This project consists of an artillery shell designed

to fight fires, strengthening the action of the firefight-

ers and Civil Protection. Its performance allows long

distance firefighting in areas of high slope and difficult

access for terrestrial means and in situations with re-

duced visibility, nighttime as well as under adverse

weather conditions.

The FIREND Project is an opportunity to ren-

der the military a new kind of missions in peacetime,

with special interest for the entire population, which

certainly contribute to raising their levels of motiva-

1

tion and to be recognized for their work performed on

behalf of the population.

This study aims to determine a geometry for a

projectile and to ensures flight stability along its tra-

jectory. This requires the use of CFD simulations,

so that the coefficients of force and moment can be

calculated, and also the pseudo-empirical prediction

methods, as is documented in external ballistics liter-

ature. As a final goal we intend to build a software

that couples results from aerodynamic analysis with

the calculation of the flight trajectory using a 6 de-

grees of freedom approach.

2 Description

For external ballistics be understood, it is neces-

sary to take into account several factors, including the

correct nomenclature [3], [4].

Three reference frames were considered in the

overall computations:

• the X Y Z referential, which is static and located

at the howitzer exit, as shown in Figure 1;

• the x’ y’ z’ referential, located in the CG of the

projectile, which is always aligned with the X Y Z

referential;

• the x y z reference frame, aligned with the body

of the projectile, and that has the same spin move-

ment as the projectile;

Figure 1: Referential used.

Using the Star−CCM+ it is possible obtain the

aerodynamic forces and moments acting on the pro-

jectile. If the gravitational force is known, the accel-

erations comes naturally from an algebraic operation.

By definition:

Fi = d (m vi)dt

(1)

and thus,

ai = dvi

dt(2)

By numerical integration, after setting ∆t as a

constant, it follows that

vinew = vi

old + ∆t.ai (3)

Integration of equation 2 gives vi = dxi/dt, which

after discretization provides us the equation of coordi-

nate position xi of the body with time. Finally xnewi

= xoldi + ∆t.vold

i gives the discrete equation of CG

position.

The rotational motion is represented with three

degrees of freedom that is three angular coordinates

which represent the body orientation in 3D space

about its mass centre: Θ = φ, θ, ψ, with φ the

spin angle, θ the pitch and ψ the yaw angle.

The orientation of the body is defined after three

sequential rotations, in the previous order φ, θ, ψ -

relatively to the referential x’ y’ z’. In order to analyze

the motion of the projectile, only the attack angle is

considered, which defined as the angle between the

axis of the projectile and the flow direction of ap-

proach in the projectile frame. With no wind, this

direction will be the direction of the trajectory.

The Euler equations of motion were used for the

3D rotation. These equations are written on a refer-

ential whose axis correspond to the principal axis of

inertia. The referential can rotate with the body to

keep unchanged the inertia moments.

The equations are:

~M = ~H + ~R−1 ~R ~H or ~M = ~I ~ΩB + ~ΩR X(~I~ΩB

)(4)

2

where

~I =

Ixx 0 0

0 Iyy 0

0 0 Izz

(5)

~Ω = (ωx;ωy;ωz) (6)

The subscripts B and R in Equation 4 represent the

body and the frame, respectively, being ~R the rotation

matrix.

The rotation matrix has 9 free variables, but con-

sidering the symmetry of the matrix, 3 of them are

automatically set. Therefore, the remaining six vari-

ables are used to define three degrees of freedom of

rotational movement in 3D. As such, it will be neces-

sary to reorthonormalize the rotation matrix.

The 6-DOF trajectory was used because it is the

model that best represents the physical phenomenon

of ballistic flight, when we want an overall assessment,

similar to what had been done in 2009 [5], when ef-

fected studies were done to calculate the trajectory

of a 155 mm projectile, resorting to the aid of the

software PRODAS R© to calculate flight behavior.

The use of CFD methodologies to approach a

problem is based on a very specific procedure. During

the pre-processing, it is necessary to define the geom-

etry of the object under study, as well as its domain,

which is divided into discrete cells (the mesh) [6]. The

choice of the mesh to be implemented is very impor-

tant because it will have a decisive role in the solution

of differential equations [7]. The CFD software allows

to calculate the forces and moments at each time and

report to the user [8], which depend on the angle of

attack [9]. It was found that the lift increases with

the angle of attack, as well as the axial force due to

the Magnus effect.

The spin stabilized projectiles for L/D ratio

greater than 7 (L is the length and D the projectile’s

caliber) tends to be unstable unless additional com-

ponents are used to ensure stability [10].

There are several factors that affect the trajectory.

They are: angle and launch speed, mass and shape of

the projectile, atmospheric wind, effect of Earth’s ro-

tation, forces and moments induced by rotation and

weather effects [3]. The velocity of the projectile has

a direct influence on the resistance it offers in flight

[11], illustrated in Figure 2.

Figure 2: Variation of drag coefficient with Mach [11].

The following set of hypotheses to simplify the en-

tire process is assumed:

• gravity constant along the trajectory of the pro-

jectile equal to 9,81 m/s2;

• the properties of the air are set for a standard

atmosphere;

• it is considered that the projectile leaves the

howitzer with an angle of attack of 0 degrees and with-

out applied forces;

• the Coriolis force was not considered due to its

low influence in short ranges;

• the surface of the projectile is considered smooth

for the purposes of friction and turbulence;

• it is considered that the CG of the projectile is

constant, as well as its moments of inertia, along the

path.

3

3 Verification and Validation

The assumptions for the projectile are:

• the howitzer induces a spinning rotation to the

projectile;

• charge 1 is used, therefore subsonic regime.

Another reason was to establish a methodology for

use in optimization processes being the CD approxi-

mately constant in the subsonic regime, having only

that the drag increases with the square of speed and

air density.

The case study for verification and validation was

based on the conventional ammunition M107, illus-

trated in Figure 3, with its main characteristics in

Table 1. These tests were carried out to load 1 (min-

imum charge of gunpowder) which corresponds to a

given rate of translation and rotation.

Figure 3: M107 ammunition geometry.

Table 1: Features with v = 211.8m/s e ω = 344 rad/s.

l (mm) m (kg) CP (mm)* CG (mm)*697.100 43.096 184.04 458.37

∗measured from the nose

The mesh is a fundamental parameter to obtaining

reliable results. After the convergence of the results is

verified for a number of cells of approximately 6.5M,

Figure 5, it was assumed that this would be the kind

of refinement required to give an answer to the pur-

pose, Figure 4.

The error obtained for the value of CD compared

with the value of CDtheoretical to 6.5M of a mesh cell,

was less than 2 %.

Figure 4: Final mesh adopted.

0,142

0,146

0,15

0,154

0,158

0,162

0 1 2 3 4 5 6 7

Cd

's v

alu

e

Number of cells in millions

Cd's convergence

Cd

Cd theoretical

Figure 5: Drag with the mesh refinement.

The physical model chosen for the CFD simula-

tions in Star−CCM+ R© was "Coupled Flow", which

is suitable for compressible flows [12]. Another ad-

vantage of this model is the robustness to solve flows

with dominant source terms because the projectile has

spin.

The turbulence model used was k − ω (SST ver-

sion) for being suitable for situations of high adverse

pressure gradient and flow rotation.

The boundary conditions of the wall were chosen

high y+ (from 30 to 150), Figure 6, to prevent the

mesh had more cells.

In Figure 6 one can observe the representation of

the y+ along the projectile. As it can be seen the

range for the values of high y+ was respected. There

have been some exceptions for which it was not possi-

ble to guarantee the assumptions made in some areas

because the flow was almost "stopped".

4

Figure 6: Representation of y+.

Simulations were performed with 0o, 2o and 4o of

angle of attack because variations of α in flight am-

munition occur mostly in this range. These values

will influence the range and stability of the projectile

along its trajectory. It is important to note that the

coefficients of forces and moments were calculated at

the CG of the projectile.

In Figure 7 can clearly be observed the influence

of the angle of attack on the projectile. For 0 angle

of attack can verify the symmetry of the turbulence

in the wake unlike what happens at 4o, noting the

variation of turbulence in the wake zone.

(a) 0o angle of attack.

(b) 4o angle of attack.

Figure 7: Variation of TKE.

To analyze the trajectory it was used two soft-

wares: Mathematica R© and PRODAS R©, which

is a recognized ballistics commercial software. In

Mathematica R© it was implemented a specific own

code, while in PRODAS R© it was integrated the CFD

results in the study of rigid body dynamics. It was

established a comparison between the values of firing

tables with the values obtained by different software

in order to verify and validate the code implemented

in Mathematica R©. The results are shown in Table

2.

Table 2: Ranges with β e v = 211.8 m/s.

βi (o) Tab (m) Math (m) PRODAS (m)29.64 3600 3549.6 3571.131.21 3700 3647.0 3669.932.99 3800 3744.3 3768.935.11 3900 3841.8 3868.137.89 4000 3939.2 3967.9

As already mentioned, the Mathematica R© was

used due to their valencies. A code was created so

that the trajectory of the projectile could be calcu-

lated at any instant of time.

In order to complement the results obtained with

PRODAS R©, a code was created, mainly because the

operating conditions of the new projectile can not be

satisfied with this software because, for example, the

future launch speed is Mach 0.3, much lower than the

typical values of speed used, and this software makes

interpolations of experimental data.

The other reason is the fact that the geome-

try can no longer be conventional and in this case,

PRODAS R© also is not suitable, although it is useful

when it is possible to apply, making the process much

quicker.

However, the code created inMathematica R© was

also verified and validated with PRODAS R©, using

for this a speed of 200 m/s and a conventional geom-

etry, which is available in the software library. Thus

it was possible to compare the results obtained with

both software. Those results are shown in Table 3.

Table 3: Comparison of the results obtained.

Variables PRODAS Mathematicavi (m/s) 200 200ω (rad/s) 324 324β (o) 30 30

xmax (m) 3236.11 3217.73tvoo (s) 19.91 19.90hmax (m) 486.16 485.7thmax

(s) 9.85 9.84

5

As it can be seen from the results presented, both

results are very similar, thus being verified and vali-

dated by the code.

4 Results

Some initial conditions that have a major role in

the design of the projectile and scope of the intended

purpose were imposed. These are shown in Table 4.

Table 4: Initial conditions of operation.

vi (m/s) m (kg)FIREND 100 10 < m < 15

The initial velocity and the total mass of the pro-

jectile have undergone significant changes. These aim

to conceive the projectile FIREND with a material

which makes the entire process more economical and

that the pressures and temperatures developed inside

the shell are much lower, so that the new material

supports them.

It were created and analyzed six geometries, vary-

ing the total length of the projectile and the shape of

his nose. The first 3 nose cone versions have a vary-

ing length, while the other have a hemispherical nose,

Table 5.

Table 5: Different geometries tested.

Nomenclature l (mm) m (kg) GSFversion 1 697.1 10.775 1.75version 2 797.1 12.371 1.19version 3 897.1 14.022 0.89version 4 697.1 10.024 1.04version 5 797.1 11.435 0.74version 6 897.1 12.792 0.58

Both parameters that were changed have the same

purpose: to increase the ability of material to be

transported inside the projectile, when compared with

conventional M107 ammunition.

The concept of hemispherical nose dates back to

the FIREND 105 mm Project. Giving this same con-

cept, applied to the same overall projectile’s length,

the carried material is to be increased to the double

of the one used on the conical profile ammunition.

In order to test and to establish a comparison be-

tween different geometries, some parameters remained

unchanged. The speed used was 100 m/s and tested

simulations into a launch angle of 60, as it is more

susceptible to the instability of the projectile.

However, it should be noted that other parameters

such as the total length, CG, CP, undergone changes

in geometry to geometry, these parameters also have a

leading role in the stability analysis of the projectile.

Based on tests performed for six different geome-

tries in order to select the geometry that best fulfilled

the requirements for the new 155 mm FIREND’s pro-

jectile, geometry version 4 was selected. It was found

that this is the most stable geometry by analysis of

the variation of the angle of attack during the trajec-

tory, compared with various geometries analyzed for

different wind speeds, and also according to the value

of the registered gyroscopic stability. The results ob-

tained are shown in Table 6.

In Table 5 it can be seen that the parameter of

gyroscopic stability is only satisfied for three of the

created versions, being version 4 one of these. This

provides a greater capacity to carry material inside

and therefore, with regard to this aspect, is the bet-

ter choice.

Table 6: Different limits as a function of wind speed forgeometry version 4.

vwind (m/s) xmax (m) hmax (m) zmax (m)0 766.99 353.79 -27.5

+20 744.94 349.62 -5.3-20 760.18 349.07 -42

6

Figure 8: Variation of α with vwind = 0 knots.

Figure 9: Variation of α with vwind = +20 knots.

Figure 10: Variation of α with vwind = -20 knots.

For this geometry the maximum angle of attack

achieved in a situation without wind, never exceeds

5.6o, while in a situation where there is the presence

of a crosswind (magnitude of 20 knots) the angle of

attack already reaches values of 10o. However, these

maximum are achieved when the projectile starts its

descending phase in the trajectory, when the travel

speed of the projectile registers its minimum, then re-

turning to low values of angle of attack. It means

that its behavior returns to a stabilize level in the

stage which is prior to his arrival at the ground. The

variation of angle of attack along the trajectory can

be seen in Figures 8, 9 and 10.

In Figure 11 one can observe the new location of

the CG and CP in geometry version 4.

Figure 11: FIREND 155 mm geometry with CP and CG.

Using the software Star − CCM+ R©, all simula-

tion methodology developed for conventional ammu-

nition M107, in the previous chapter, was applied in

this new configuration of the projectile, i.e., in the

FIREND 155mm geometry. Thus, the final mesh got

approximately 7M cells, present in Figure 12, like the

mesh implemented for the M107 ammunition in the

previous section. Also the parameters implemented

in Star − CCM+ R© were the same.

Figure 12: Mesh used in FIREND 155 mm.

With the results of the aerodynamic characteris-

tics for the three angles of attack analyzed (0o, 2o

and 4o), under the imposed conditions, the curves

of forces and moments acting on the projectile were

traced in order to observe the tendency. These results

are shown in Tables 7 and 8 and the characteristics

curves are shown in Figures 13 and 14. It is important

to note that the moments were calculated at the CG

of the projectile.

Table 7: Results of forces with α.

α (o) Magnus (N) Lift (N) Drag (N)0 -0.173 0.011 -17.0442 0.192 7.106 -17.8364 0.706 13.616 -19.173

7

Table 8: Results of moments in the CG with α.

α (o) Pitch (N.m) Yaw (N.m) Roll (N.m)0 0.003 0.043 -0.0622 -2.154 0.043 -0.0624 -4.411 0.013 -0.062

As can be seen in Figure 13, increasing the angle

of attack causes large changes in the lift and in partic-

ular, Figure 14, in pitch, by varying these parameters

in an approximately linear way. In the case of the lift,

which for 0 angle of attack was virtually nil, at 4o

shall be of considerable value. The same goes with

the pitch, resulting of pressure distribution along the

projectile.

y = -0,068x2 - 0,2602x - 17,044

y = -0,0747x2 + 3,7025x

y = 0,0403x2 + 0,0153x

-25

-15

-5

5

15

0 1 2 3 4

Forc

es

(N)

Attack angle (°)

Drag

Lift

Magnus

Figure 13: Variation of the forces with α.

y = -0,0127x2 - 1,0517x

y = -0,009x2 + 0,0393x

y = 5E-06x2 + 3E-05x - 0,0624

-5

-4

-3

-2

-1

0

1

0 1 2 3 4

Mo

me

nts

(N

.m)

Attack angle (°)

Pitch

Yaw

Roll

Figure 14: Variation of the moments with α.

Similar to the analysis performed for the trajec-

tory of the projectile with 60o of launch angle, after

having defined the final geometry to adopt, additional

tests were also conducted for other launch angles, es-

pecially for 30o and 45o. As was done previously, it

was now found the influence of wind at 20 knots in

projectile either in the positive direction, but also in

the negative sense. The results are shown in Table 9.

Table 9: Limits to wind for FIREND 155 mm with β =60o.

vwind (m/s) xmax (m) hmax (m) zmax (m)0 766.99 353.79 -27.5

+20 744.94 349.62 -5.3-20 760.18 349.07 -42

5 Conclusions

The main conclusions are the following:

• The value of the forces and moments acting on

the projectile depend significantly on the angle of at-

tack. For α = 0o all moments and forces are practi-

cally inexistent, except drag and roll;

• The maximum angle of attack reached by the

projectile along its trajectory is directly dependent

on the launch angle;

• It was found that it would not be possible to

increase the length of the projectile, ensuring its sta-

bility in flight as described in the literature.

• For the initial conditions imposed for the release

of the projectile, the hemispherical profile is more sta-

ble than the conical profile. With this new profile in

addition to respect the stability parameter, also sig-

nificantly increases the amount of load to be carried

inside the projectile;

• Were also carried out simulations taking into ac-

count the wind speed and it was concluded that the

most critical situation in order to unsettle the flight of

the projectile is to crosswind with positive direction.

It was shown that the 155 mm projectile FIREND

would be more likely to unsettle when subjected to

wind than the M107;

• It was also concluded that the lateral deviation

suffered by the projectile has a strong dependence on

8

the time of pitch, and the higher the value the first

derived of this moment, lower the amount of lateral

deviation of the projectile. It also presents some in-

fluence in variation of the angle of attack so that the

larger the lateral deviation greater will be the varia-

tion of angle of attack.

• A process was established, verified and validated,

which allows to characterize in a structured and se-

quential way the specificities of a munition with re-

gard to external ballistics. This methodology will

be of great importance in future developments of the

project. It should be noted that even if this form is

considered, many other aspects such as the interior of

munition or the material which is manufactured have

implications in the mass, in moments of inertia, in

CG, among others.

• According to the methodology and parameters

defined, it’s expectable to obtain aerodynamic and

external ballistics results lower than 2%.

References

[1] F. Lima, “Desenvolvimento de munições de artil-

haria para combate a incêndios florestais,” Mas-

ter’s thesis, Lisboa: Instituto Superior Técnico,

2005.

[2] J. Calado, “Projeto de um projétil de calibre

155mm para combate a incêndios florestais,”

Master’s thesis, Lisboa: Instituto Superior Téc-

nico, 2013.

[3] D. Carlucci and S. Jacobson, Ballistics: Theory

and Design of Guns and Ammunition. Taylor

& Francis, 2007.

[4] E. Dias, “Desenvolvimento do conceito, fabrico

e ensaio de um projéctil mecânico para combate

a incêndios florestais: Firend.” Master’s thesis,

Lisboa: Instituto Superior Técnico, 2007.

[5] M. Khalil, “Dispersion analysis for spinning ar-

tillery projectile,” in ASAT-13FM-03, 13th In-

ternational conf. Aerospace Science & Aviation

Technology.

[6] R. Barata, “An approach to ventilation design

application to loureshopping case,” Master’s the-

sis, Lisboa: Instituto Superior Técnico, 2009.

[7] A. S. Vianna Jr, F. de Siqueira, L. R. Quintal,

and A. L. S. Lima, “CFD e a defesa química,”

Revista Virtual de Química, vol. 6, no. 3, pp.

815–832, 2014.

[8] Advanced Coupled CFD/RBD Calculations of

Free-Flight Projectile Aerodynamics, 2005.

[9] J. Garibaldi, M. Storti, L. Battaglia, and

J. D’Elia, “Numerical simulations of the flow

around,” 2007.

[10] BlueCAPE, “Trajectory analysis of a spinning

projectile,” 2007.

[11] R. McCoy, Modern Exterior Ballistics: The

Launch and Flight Dynamics of Symmetric Pro-

jectiles, ser. Schiffer military history. Schiffer

Publishing, Limited, 2012.

[12] “Star-CCM+, User Guide, version 8.02, CD-

Adapco,” 2013.

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